# KC-space

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of Hausdorffness

View other variations of Hausdorffness | read a survey article on varying Hausdorffness

## Contents

## Definition

### Symbol-free definition

A topological space is termed a **KC-space** if every compact subset of it is closed (here, by *compact subset*, we mean a subset which is a compact space under the subspace topology).

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Hausdorff space | any two distinct points can be separated by disjoint open subsets | Hausdorff implies KC | KC not implies Hausdorff | Weakly Hausdorff space|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

weakly Hausdorff space | any subset of it arising as the continuous image of a compact Hausdorff space is closed in it | | | ||

US-space | KC implies US | US not implies KC | | | |

T1 space | points are closed | KC implies T1 | T1 not implies KC | | |

Kolmogorov space | any two points can be distinguished | (via T1) | (via T1) | | |